Well, for unlocking the vehicles with the door handles(Passive Entry) this would in theory work. However, this would not allow you to start the vehicle. BMW uses peripheral antennas for transmitter function relay only, not for actual locking function communications.
To elaborate, when you come up to your car with a comfort access key the antenna in the door handle you are approaching(peripheral antenna) recognizes the key via the AM signal the OP was referring to(125kHz). This wakes the key and requests an ID response on a different frequency(315MHz), which is sent by the key to the regular keyless entry antenna(FBD). So, as long as the key is within range of the car(if you hit the button on the remote, does it work?), this "hack" should in theory work to unlock the vehicle.
However, Passive Go(Keyless Starting) works slightly differently. When a start command is initiated inside the vehicle by pushing the button, it uses the comfort access aerials inside the vehicle(separate from the peripheral antennas) to identify the key via low power transmissions. The interior antenna still tells the key(on the 125kHz band) to transmit an ID packet, but it also instructs it to includes immobilizer data and as such is transmitted at a much lower power by the key(still in the 315MHz range). This distinction allows the vehicle to know if the key is inside or outside the vehicle, useful for determining if the doors should be allowed to lock(accidental locking prevention). After all, why do we need to use a full strength signal if the key is supposed to be within 10 feet of the FBD prior to starting?
This means the key's responding signals would have to be amplified as well to implement this "hack" to start the car, which is much less probable. Is the thief going to slip an amplifier into your purse when you are not looking, so they can steal your car? They may as well just palm your key if they are going that route...